Question: $ A = \left[\begin{array}{rr}4 & 5 \\ -1 & -1\end{array}\right]$ $ B = \left[\begin{array}{rrr}1 & -2 & 3 \\ -2 & 4 & 3\end{array}\right]$ What is $ A B$ ?
Answer: Because $ A$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A B = \left[\begin{array}{rr}{4} & {5} \\ {-1} & {-1}\end{array}\right] \left[\begin{array}{rrr}{1} & \color{#DF0030}{-2} & \color{#9D38BD}{3} \\ {-2} & \color{#DF0030}{4} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{4}\cdot{1}+{5}\cdot{-2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{4}\cdot{1}+{5}\cdot{-2} & ? & ? \\ {-1}\cdot{1}+{-1}\cdot{-2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{4}\cdot{1}+{5}\cdot{-2} & {4}\cdot\color{#DF0030}{-2}+{5}\cdot\color{#DF0030}{4} & ? \\ {-1}\cdot{1}+{-1}\cdot{-2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{4}\cdot{1}+{5}\cdot{-2} & {4}\cdot\color{#DF0030}{-2}+{5}\cdot\color{#DF0030}{4} & {4}\cdot\color{#9D38BD}{3}+{5}\cdot\color{#9D38BD}{3} \\ {-1}\cdot{1}+{-1}\cdot{-2} & {-1}\cdot\color{#DF0030}{-2}+{-1}\cdot\color{#DF0030}{4} & {-1}\cdot\color{#9D38BD}{3}+{-1}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-6 & 12 & 27 \\ 1 & -2 & -6\end{array}\right] $